t2solf05 - STA 302 / 1001 F - Fall 2005 Test 2 November 16,...

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Unformatted text preview: STA 302 / 1001 F - Fall 2005 Test 2 November 16, 2005 LAST NAME: SOLUTIONS FIRST NAME: STUDENT NUMBER: ENROLLED IN: (circle one) STA 302 STA 1001 INSTRUCTIONS: Time: 90 minutes Aids allowed: calculator. A table of values from the t distribution is on the second to last page (page 9). A table of formulae is on the last page (page 10). For all questions you can assume that the formulae on page 10 are known. Total points: 44 1 2ab 2cd 3abc 3d 3efg 4 7 7 10 3 9 1 1. (4 points) Suppose that X is a 2 1 random vector with E( X ) = 1 2 ! and Cov( X ) = 4- 1- 1 9 ! . Y is another random vector with Y = AX where A is the constant matrix A = 1-2 3 ! . Find the expectation of Y and the variance-covariance matrix for Y . E( Y ) = A E( X ) = 1-2 3 ! 1 2 ! =-3 6 ! Cov( Y ) = A Cov( X ) A = 1-2 3 ! 4-1-1 9 ! 1-2 3 ! = 6-19-3 27 ! 1-2 3 ! = 44-57-57 81 ! 2 2. (14 points) (a) Write the simple linear regression model in matrix terms, defining all terms. (3 marks) Y = X + where Y = Y 1 Y 2 . . . Y n = 1 ! X = 1 X 1 1 X 2 . . . . . . 1 X n = 1 2 . . . n (b) Explain why Cov( ) = 2 I follows from the assumptions of simple linear regression. (4 marks) Cov ( ) is the n n matrix with i th diagonal entry equal to the variance of i and ij th off- diagonal entry equal to Cov ( i , j ) . Since two regression assumptions are that Var ( i ) = 2 for all i and Cov ( i , j ) = 0 , it follows that Cov ( ) = 2 I ....
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t2solf05 - STA 302 / 1001 F - Fall 2005 Test 2 November 16,...

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